Algorithms Seminar

How can you construct a mosaic of tiles, where each fragment is
repeated infinitely many times and yet no single fragment is sufficient
to determine the structure of the whole mosaic? In mathematical
crystallography, such aperiodic tilings are studied in the theory
of quasicrystals. Regular crystals are defined by translational
symmetry, whereas the self-similar structure of quasicrystals
is characterized by inflational symmetry. The development of the
theory of quasicrystals ensued from a revolution in solid state
physics, due to the 1984 discovery of metallic alloys which exhibited
a long-range atomic order with five-fold rotational symmetry,
a structure that is fundamentally incompatible with periodic tilings,
which forced a reconsideration of the definition of a crystal.

An aperiodic tiling can be generated by the Voronoi diagram
of an aperiodic Delone point set. Such point sets may be constructed
using the cut and project method, where a 2N dimensional periodic
lattice is projected onto an N dimensional plane which is irrationally
oriented with respect to the lattice. Hence, the coordinates of
these quasicrystal points need to involve only integers and an
irrational number, typically the Golden Ratio. It turns out that
these N dimensional quasicrystals may be studied as lattices of
one-dimensional quasicrystals since the points that lie on any
straight line through an N dimensional quasicrystal correspond
to a one-dimensional quasicrystal. Typically, a one-dimensional
quasicrystal is composed of only three distinct tiles. Hence,
it is easy to generate quasicrystal points using an iterative
numerical algorithm. However, a more robust alternative is to
exploit the self-similar structure of a quasicrystal, viewing
it as the fixed point of a set of substitution rules that act
recursively on a finite alphabet of possible tile arrangements.
Finally, in order to efficiently render quasicrystals on a computer
screen, instead of calculating an exact Voronoi diagram, it is
possible to use a discrete Voronoi diagram since there is a minimum
distance between any two quasicrystal points.

This seminar, based on the research of Prof. Jiri Patera and
his group at Centre de Recherches Mathematiques, will present
the mathematical basis for these methods along with a view towards
possible applications in computer science, ranging from random
number generators to texture synthesis.